STAN Examples
Songpeng Zu
20 October 2020
Set STAN Environment
install.packages("cmdstanr", repos = c("https://mc-stan.org/r-packages/", getOption("repos")))
## The two are used for STAN result analysis.
install.packages("bayesplot")
install.packages("posterior")Install cmdstan either from the cmdstan repository or by cmdstanr
Set the basic environment
library(cmdstanr)
library(bayesplot)
library(posterior)
## for bayesplot plotting
color_scheme_set("brewer-Spectral")
## If you use cmdstanr install cmdstan, you can ignore this.
set_cmdstan_path(path = paste(Sys.getenv("HOME"),
"softwares",
"cmdstan-2.23.0", sep = "/"))
## set the parameter for STAN sampling with multi-core support.
options(mc.cores = 3)Example 1: Multivariate normal with missing data
Data
Our observations are from \(\mathcal{N}(0, \Sigma)\), where \(\Sigma\) is unknown.
\[\begin{pmatrix} 1 & -1 & -1 & 1 & 2 & 2 & -2 & -2 & * & * & * & *\\ 1 & -1 & 1 & -1 & * & * & * & * & 2 & 2 & -2 & -2 \end{pmatrix}\]Each column corresponds to one sample, and \(*\) are missing data.
How we infer \(\Sigma\) using STAN
- load the model
camel_mdl <- cmdstan_model("camel/camel2.stan", compile = TRUE)## Compiling Stan program...camel_mdl$print()## /*
## * Camel: Multivariate normal with structured missing data
## * http://www.openbugs.net/Examples/Camel.html
## *
## */
## transformed data {
## vector[2] Y[4];
## real Y1[4]; // missing y2
## real Y2[4]; // msising y1
##
## matrix[2, 2] S;
##
## S[1, 1] = 1;
## S[1, 2] = 0;
## S[2, 1] = 0;
## S[2, 2] = 1;
##
## Y[1, 1] = 1.;
## Y[1, 2] = 1.;
## Y[2, 1] = 1.;
## Y[2, 2] = -1.;
## Y[3, 1] = -1.;
## Y[3, 2] = 1.;
## Y[4, 1] = -1.;
## Y[4, 2] = -1.;
##
## Y1[1] = 2.;
## Y1[2] = 2.;
## Y1[3] = -2.;
## Y1[4] = -2.;
##
## Y2[1] = 2.;
## Y2[2] = 2.;
## Y2[3] = -2.;
## Y2[4] = -2.;
## vector[2] mu = rep_vector(0.0, 2);
## }
##
##
## parameters {
## // vector[2] mu;
## cov_matrix[2] Sigma;
## }
##
## model {
## // mu ~ normal(0, 10);
## // Sigma ~ inv_wishart(3, S);
## for (n in 1:4) {
## Y[n] ~ multi_normal(mu, Sigma);
## }
## Y1 ~ normal(mu[1], sqrt(Sigma[1, 1]));
## Y2 ~ normal(mu[2], sqrt(Sigma[2, 2]));
##
## // Use Jeffery's prior: p(Sigma) is proportional to |Sigma|^{-3/2}
##
## // `target` is the conserved word in STAN.
## // It adds all the log density function, and it (or you)
## // can ignore the normalizing constants as long as the constants have no
## // influence for getting the gradients.
## target += -1.5 * log(determinant(Sigma));
## }
##
## generated quantities {
## real<lower=-1,upper= 1> rho;
## rho = Sigma[1, 2] / sqrt(Sigma[1, 1] * Sigma[2, 2]);
##
## real y2condy1[4];
## real y1condy2[4];
## real muy2condy1[4];
## real muy1condy2[4];
##
## for (i in 1:4) {
## // posterior mean
## muy2condy1[i] = mu[2] + sqrt(Sigma[2,2]) * rho * (Y1[i] - mu[1])/sqrt(Sigma[1,1]);
## // posterior sampling from a conditional normal distribution.
## y2condy1[i] = normal_rng(muy2condy1[i],
## sqrt((1-rho*rho) * Sigma[2,2]));
## }
##
## for (i in 1:4) {
## muy1condy2[i] = mu[1] + sqrt(Sigma[1, 1]) * rho * (Y2[i]- mu[2]) / sqrt(Sigma[2,2]);
## y1condy2[i] = normal_rng(muy1condy2[i],
## sqrt((1-rho*rho) * Sigma[1,1]));
## }
## }- sampling the parameter
camel_sampler <- camel_mdl$sample(data = NULL,
seed = 355113,
parallel_chains = 4,
show_message = TRUE,
## increase adapt delta, default is 0.80
adapt_delta = 0.85,
iter_sampling = 1000
## output_dir = "./"
)## Running MCMC with 4 parallel chains...
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##
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## Total execution time: 0.7 seconds.camel_draws <- camel_sampler$draws()- show the posterior results
bayesplot::mcmc_combo(camel_draws,
pars = c("rho"), widths = c(1, 3),
combo = c("dens_overlay", "trace")
)
Example 2: Finite normal mixture model
load STAN script for both simulation and inference
mixture_model <- cmdstan_model("mixture/normal_mixture_k_order.stan",
compile = TRUE
)## Compiling Stan program...Generate the simulation data
N <- 1200
N1 <- 400
N2 <- 400
musig <- 10.0
alpha <- c(1.0, 1.0, 1.0)
sigma <- c(1.0, 1.0, 1.0)
sim_sampler <- mixture_model$sample(
data = list(
K = 3,
N = N,
N1 = N1,
N2 = N2,
y = rep(1, N),
musig = musig,
alpha = alpha,
sigma = sigma
),
seed = 355113,
chains = 1,
iter_sampling = 1,
parallel_chains = getOption("mc.cores", 1),
show_message = TRUE,
init = list(
list(
theta = c(N1 / N, N2 / N, 1 - N1 / N - N2 / N),
mu = c(-5.0, 0.0, 5)
## sigma = c(0.5, 1, 1)
)
),
fixed_param = TRUE
)## Running MCMC with 1 chain...
##
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## Chain 1 finished in 0.1 seconds.sim_draws <- sim_sampler$draws()
## dimnames(sim_draws)$variable[1:10]
gy <- c(sim_draws[1, 1, grep("gy", dimnames(sim_draws)$variable)])
hist(gy, breaks = 50)
Sampling with STAN
sim_sampler <- mixture_model$sample(
data = list(
K = 3,
N = N,
N1 = N1,
N2 = N2,
y = gy,
musig = musig,
alpha = alpha,
sigma = sigma
),
seed = 355113,
chains = 4,
iter_sampling = 1000,
fixed_param = FALSE,
## init = list(
## list(
## theta = c(0.4, 0.3, 0.3),
## mu = c(-4.0, 0.0, 4),
## sigma = c(10, 10, 10)
## ),
## list(
## theta = c(0.2, 0.3, 0.5),
## mu = c(-10.0, 0.0, 10),
## sigma = c(10.0, 10, 10.0)
## )
## ),
parallel_chains = getOption("mc.cores", 1),
## output_dir = "./",
show_message = TRUE
)## Running MCMC with 4 chains, at most 3 in parallel...
##
## Chain 1 Iteration: 1 / 2000 [ 0%] (Warmup)## Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:## Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/jv/thfh42cj1zv2k4csp7sk7qg40000gn/T/RtmpVbfwsj/model-1057c5ea166f9.stan', line 28, column 3 to column 49)## Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,## Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.## Chain 1## Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:## Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/jv/thfh42cj1zv2k4csp7sk7qg40000gn/T/RtmpVbfwsj/model-1057c5ea166f9.stan', line 28, column 3 to column 49)## Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,## Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.## Chain 1## Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:## Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/jv/thfh42cj1zv2k4csp7sk7qg40000gn/T/RtmpVbfwsj/model-1057c5ea166f9.stan', line 28, column 3 to column 49)## Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,## Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.## Chain 1## Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:## Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/jv/thfh42cj1zv2k4csp7sk7qg40000gn/T/RtmpVbfwsj/model-1057c5ea166f9.stan', line 28, column 3 to column 49)## Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,## Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.## Chain 1## Chain 2 Iteration: 1 / 2000 [ 0%] (Warmup)## Chain 2 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:## Chain 2 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/jv/thfh42cj1zv2k4csp7sk7qg40000gn/T/RtmpVbfwsj/model-1057c5ea166f9.stan', line 28, column 3 to column 49)## Chain 2 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,## Chain 2 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.## Chain 2## Chain 3 Iteration: 1 / 2000 [ 0%] (Warmup)## Chain 3 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:## Chain 3 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/jv/thfh42cj1zv2k4csp7sk7qg40000gn/T/RtmpVbfwsj/model-1057c5ea166f9.stan', line 28, column 3 to column 49)## Chain 3 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,## Chain 3 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.## Chain 3## Chain 3 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:## Chain 3 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/jv/thfh42cj1zv2k4csp7sk7qg40000gn/T/RtmpVbfwsj/model-1057c5ea166f9.stan', line 28, column 3 to column 49)## Chain 3 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,## Chain 3 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.## Chain 3## Chain 2 Iteration: 100 / 2000 [ 5%] (Warmup)
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##
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## Total execution time: 19.6 seconds.View the results.
ey <- sim_sampler$draws()
bayesplot::mcmc_combo(ey[, , grep("mu", dimnames(ey)$variable)],
widths = c(1, 3),
combo = c("dens_overlay", "trace"))
If we set unordered \(mu\) in STAN
uy <- umixture_sampler$draws()
bayesplot::mcmc_combo(uy[, , grep("mu", dimnames(uy)$variable)],
widths = c(1, 3),
combo = c("dens_overlay", "trace"))